### Hypothesis Testing Examples
### Locating the critical value of the test.

#######################################################
#######################################################
# Example 1: Normal assume theta0 < theta1

# How to make a normal table in R/S-Plus

#######################################################
# Suppose 

alpha = 0.05

theta0 = 25; sigma0 = 3; n = 30;

xx = seq(theta0-5*sigma0/sqrt(n),theta0+5*sigma0/sqrt(n),.01)

yy = dnorm(xx, mean = theta0, sd = sigma0/sqrt(n))

plot(xx,yy,type="l")  # plot the null distribution

#######################################################

low = 1.6  # change these values to locate alpha = 0.05
up = 1.7

k = seq(low,up,0.001)

cv = 1 - pnorm(k)  # critical value is for alpha = 0.05
cv = round(cv,4)

cv = matrix(c(k,cv),ncol=2)
cv

cv = cv[cv[,2]==alpha]

rr = theta0 + cv[1]*sigma0/sqrt(n)
rr

#######################################################
#######################################################
# Example 2: Exponential assume lambda0 > lambda1

#######################################################
# Suppose 

alpha = 0.05

lambda0 = 1.3; n = 3;

m = n/lambda0; v = n/lambda0^n;

xx = seq(0,m+5*v,.01)

xx = seq(0,5,0.1)

yy = dgamma(xx, shape = n, rate = lambda0)

plot(xx,yy,type="l")

#######################################################

low = 0  # change these values to locate alpha = 0.05
up = 5

k = seq(low, up, 0.001)

cv = 1 - pgamma(k,shape = n, rate = lambda0)  # critical value is for alpha = 0.05
cv = round(cv,4)

cv = matrix(c(k,cv),ncol=2)
cv

cv = cv[cv[,2]==alpha]

rr = cv[1]
rr

#######################################################
#######################################################
# Example 3: Poisson assume lambda0 > lambda1

#######################################################
# Suppose 

alpha = 0.05

lambda0 = 1; n = 10;

xx = seq(0,3*n*lambda0,1)

yy = dpois(xx, lambda = n*lambda0)

plot(xx,yy,type="h")  # plot the null distribution

#######################################################

low = 0  # change these values to locate alpha = 0.05
up = 30

k = seq(low, up, 1)

cv = ppois(k,n*lambda0)  # critical value is for alpha = 0.05
cv = round(cv,4)

cv = matrix(c(k,cv),ncol=2)
cv

cv = matrix(cv[cv[,2]<=alpha],ncol=2)

row.max = dim(cv)[1]  # Find the last row
cv = cv[row.max,]

rr = cv[1]
rr					# Note: This is a concervative test and that the alpha here is less than alpha = 0.05

#######################################################
#######################################################
# Example 4: Bernoulli assume p0 < p1

#######################################################
# Suppose 

alpha = 0.05

p0 = .5; n = 20;

xx = seq(0,n,1)

yy = dbinom(xx, n, p0)

plot(xx,yy,type="h")  # plot the null distribution

#######################################################

low = 0  # change these values to locate alpha = 0.05
up = n

k = seq(low, up, 1)

cv = 1 - pbinom(k,n,p0)  # critical value is for alpha = 0.05
cv = round(cv,4)

cv = matrix(c(k,cv),ncol=2)
cv

cv = matrix(cv[cv[,2]<=alpha],ncol=2)

cv = cv[1,]

rr = cv[1]
rr					# Note: This is a concervative test and that the alpha here is less than alpha = 0.05